about law of total probability and conditional probability

Question

I have a basic question about law of total probability and conditional probability. We know from conditional probability that $ P[A \cap B] = P[A|B] P[B] = P[B|A] P[A] $ (1)
We also know from law of total probability that $ P[B] = \sum\limits_{i=1}^N P[A_i \cap B] = \sum\limits_{i=1}^N P[B \cap A_i] $ (2)
If we combine the two, I think we can come up with
$ P[B] = \sum\limits_{i=1}^N P[A_i|B] P[B] = \sum\limits_{i=1}^N P[B|A_i] P[A_i] $ (1), (2)
$ P[B] = \sum\limits_{i=1}^N P[A_i|B] P[B] $
The part above disturbs me. It seems something is wrong with it but I can't understand what.
Someone to help me clarify this?
Thanks in advance.

Answer 1

There is nothing wrong. Note that the formula $$\sum_{i=1}^N P(A_i |B)P(B)= P(B)$$ is equivalent to $$\sum_{i=1}^N P(A_i |B)= 1.$$ The last formula is just a consequence of the fact, that the mapping $A \mapsto P(A|B)$ is in fact a probability measure, and thus if $S$ is the entire sample space, then we do have that $P(S|B)=1$.